Problem: Simplify; express your answer in exponential form. Assume $k\neq 0, p\neq 0$. $\dfrac{{(k^{4}p^{5})^{-3}}}{{(k^{-5}p^{-5})^{-5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(k^{4}p^{5})^{-3} = (k^{4})^{-3}(p^{5})^{-3}}$ On the left, we have ${k^{4}}$ to the exponent ${-3}$ . Now ${4 \times -3 = -12}$ , so ${(k^{4})^{-3} = k^{-12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(k^{4}p^{5})^{-3}}}{{(k^{-5}p^{-5})^{-5}}} = \dfrac{{k^{-12}p^{-15}}}{{k^{25}p^{25}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{-12}p^{-15}}}{{k^{25}p^{25}}} = \dfrac{{k^{-12}}}{{k^{25}}} \cdot \dfrac{{p^{-15}}}{{p^{25}}} = k^{{-12} - {25}} \cdot p^{{-15} - {25}} = k^{-37}p^{-40}$